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Mathematics > Combinatorics

Title:
Discreet Coin Weighings and the Sorting Strategy

Abstract: In 2007, Alexander Shapovalov posed an old twist on the classical coin
weighing problem by asking for strategies that manage to conceal the identities
of specific coins while providing general information on the number of fake
coins. In 2015, Diaco and Khovanova studied various cases of these "discreet
strategies" and introduced the revealing factor, a measure of the information
that is revealed.
In this paper we discuss a natural coin weighing strategy which we call the
sorting strategy: divide the coins into equal piles and sort them by weight. We
study the instances when the strategy is discreet, and given an outcome of the
sorting strategy, the possible number of fake coins. We prove that in many
cases, the number of fake coins can be any value in an arithmetic progression
whose length depends linearly on the number of coins in each pile. We also show
the strategy can be discreet when the number of fake coins is any value within
an arithmetic subsequence whose length also depends linearly on the number of
coins in each pile. We arrive at these results by connecting our work to the
classic Frobenius coin problem. In addition, we calculate the revealing factor
for the sorting strategy.